Spectral investigation of annular flow

Abstract

This graduate thesis has been devoted to the examination of spectral methods. The object of the research is to analyse the accuracy and precision of different numerical schemes developed on the basis of mathematical models for the Poisson and biharmonic equations, as well as to evaluate their capacity to produce reliable and consistent results in the context of solving a problem of the motion of a single-phase fluid in the region under consideration. The region under consideration has been defined by annular geometry as follows: The two-dimensional case has been defined by the geometry of an annulus with an inner radius of r0 and an outer radius of r1; The three-dimensional case has been defined by the geometry of inner and outer surfaces of two cylindrical tubes inserted one into the other, where the radius of the inner cylinder is given by r0 and the radius of the outer cylinder is given by r1. Of all the spectral methods, only two basic methods have been considered in detail in the thesis: the Galerkin method and the collocation method. As the basis function we have chosen the orthogonal Legendre and Chebyshev polynomials for a bounded non-periodic interval in the radial direction and the Fourier series for periodic intervals in the angular and z coordinate directions. The Galerkin and collocation methods have been used to approximate the Poisson and biharmonic equations. The relevant algorithms have been worked out to create numerical schemes in the Python programming language and to implement them in computer software. The research on the Poisson and biharmonic equations has helped to elaborate a new technique for discretising the Navier-Stokes equations in a three-dimensional space. Here it is worth noting that this is a new and to some extent unique material, because for problems such as the motion of a single-phase fluid in annular geometry there have not been any algorithms designed on the basis of spectral methods until now. The analysis of the numerical results has illustrated that both methods show good spectral convergence. However, a detailed investigation has also been carried out to clarify the extent to which both methods produce consistent results in the presence of noise caused by different errors. The correlation between the condition number of matrices and the number of quadrature points has also been studied in detail in this way. Further analysis of this particular issue has shown that when using the collocation method, the matrix of the biharmonic equation is affected by round-off-error-dependent distortions. The research has demonstrated that the Galerkin methods for both the Poisson and biharmonic equations in annular geometry are reliable and consistent. Therefore, it would seem reasonable to conclude that the Galerkin methods hold significant potential for investigating more complex problems.